Densities for Elliptic Curves over Global Function Fields
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866908645554388992 |
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| author | Yao, Andrew |
| author_facet | Yao, Andrew |
| contents | Let $K$ be a global function field. We obtain a set of formulas for the densities of the Kodaira types and Tamagawa numbers of elliptic curves over a completion of $K$ that is independent of the field's characteristic. Furthermore, for a finite field $F$ and real numbers $s$ and $ε$ such that $s>1$ and $ε>0$, we prove that there exists a global function field $K$ such that the full constant field of $K$ is $F$ and the value of the zeta function of $K$ at $s$ is less than $1+ε$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_11437 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Densities for Elliptic Curves over Global Function Fields Yao, Andrew Number Theory Algebraic Geometry Let $K$ be a global function field. We obtain a set of formulas for the densities of the Kodaira types and Tamagawa numbers of elliptic curves over a completion of $K$ that is independent of the field's characteristic. Furthermore, for a finite field $F$ and real numbers $s$ and $ε$ such that $s>1$ and $ε>0$, we prove that there exists a global function field $K$ such that the full constant field of $K$ is $F$ and the value of the zeta function of $K$ at $s$ is less than $1+ε$. |
| title | Densities for Elliptic Curves over Global Function Fields |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2301.11437 |